A theorem on the multiplicity of the singular spectrum of a general Anderson-type Hamiltonian

Abstract

In this work, we study the multiplicity of the singular spectrum for operators of the form $A^{\omega }=A+{\sum }_n{\omega }_n{C}_n$ on a separable Hilbert space $\mathcal{H}$, where $A$ is a self-adjoint operator and $\{C_n{\}}_n$ is a countable collection of non-negative finite-rank operators. When $\{{\omega }_n{\}}_n$ are independent real random variables with absolutely continuous distributions, we show that the multiplicity of the singular spectrum is almost surely bounded above by the maximum algebraic multiplicity of the eigenvalues of the operator $\sqrt{C_n}(A^{\omega }-z{)}^{-1}\sqrt{C_n}$ for all $n$ and almost all $(z,\omega )$. The result is optimal in the sense that there are operators for which the bound is achieved. We also provide an effective bound on the multiplicity of the singular spectrum for some special cases.